So I'm just a completionist and my buddy told me after he beat the main story he went to Diamond City and saw quite the mishap which started the quest. I too have beat the main story and im a level 55 but I have no luck starting that quest. Is there anything I need to do first or what?

The quest starts the first time you enter Diamond City after reaching level 35; has nothing to do with beating the main quest. It's possible that you just ignored it or missed it when it happened, I guess. It could also be possible that some of your story options in the MQ stopped it from happening, I suppose, but that seems unlikely.

Maybe try taking Piper with you to Diamond City? Moving her to another location before the quest should have been triggered might have stopped it from working. I would also try fast traveling close to Diamond City and walking in rather than fast traveling right into it; I've seen that screw up quests triggering a couple of times in Skyrim before, so it's worth a shot, I guess.